90 research outputs found
Colorful Borsuk--Ulam theorems and applications
We prove a colorful generalization of the Borsuk--Ulam theorem and derive
colorful consequences from it, such as a colorful generalization of the ham
sandwich theorem. Even in the uncolored case this specializes to a
strengthening of the ham sandwich theorem, which given an additional condition,
contains a result of B\'{a}r\'{a}ny, Hubard, and Jer\'{o}nimo on well-separated
measures as a special case. We prove a colorful generalization of Fan's
antipodal sphere covering theorem, we derive a short proof of Gale's colorful
KKM theorem, and we prove a colorful generalization of Brouwer's fixed point
theorem. Our results also provide an alternative between Radon-type
intersection results and KKM-type covering results. Finally, we prove colorful
Borsuk--Ulam theorems for higher symmetry.Comment: 15 page
The topological Tverberg problem beyond prime powers
Tverberg-type theory aims to establish sufficient conditions for a simplicial
complex such that every continuous map maps points from pairwise disjoint faces to the same point in
. Such results are plentiful for a power of a prime. However,
for with at least two distinct prime divisors, results that guarantee the
existence of -fold points of coincidence are non-existent---aside from
immediate corollaries of the prime power case. Here we present a general method
that yields such results beyond the case of prime powers. In particular, we
prove previously conjectured upper bounds for the topological Tverberg problem
for all .Comment: 13 pages, 2 figure
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